Extension and restriction

An extension of a temperament on a JI subgroup G to a larger subgroup G' is a new temperament that contains the same intervals as the original temperament (in particular the rank remains the same), with the same JI interpretations in the subgroup G, but gives the same intervals new JI interpretations not in the original subgroup (but are in the larger subgroup).

For example, septimal meantone and flattone are both extensions of 5-limit (2.3.5) meantone to the 7-limit (2.3.5.7), because C-E (4 fifths) represents 5/4 in both. They are different extensions, because in septimal meantone, 7/4 is C-A# (10 fifths), while in flattone, 7/4 is C-Bbb (-9 fifths).